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I have to minimize $f(x)=x^4-24x^2$ starting on the point $x_o=1$. The method converge to $x=0$, but i know that the solution is $x=+-2\sqrt{3}$. The hessian and the derivate of the function are $C_2$-smooth. $H[f(x)]^{-1}=\frac{1}{12x^2-48}$.

The method that i am using is: $x_{k+1} = x_k - H[f(x_k)]^{-1}*\nabla f(x_k)$.

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    $\begingroup$ Sure. The method converges to one of the zeroes of the derivative and does it pretty fast. Nobody has ever said that it should converge to the zero you really want. If you want to enforce the choice, couple it with bisection or use some other technique. $\endgroup$
    – fedja
    Commented Jun 27, 2019 at 21:22

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Newton's is not really a "minimizing method". If you're using Newton's method to find a root of $f'$, the root you find might be a local minimum, local maximum or neither. To remove the root $x=0$ from consideration, you might try finding a root of $f'(x)/x$ instead.

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A serious Newton minimization algorithm, sometimes called modified Newton algorithm, will employ safeguarding in the form of line search or trust region, to enforce descent across (major) iterations. Such safeguarding is necessary, even for convex problems, to ensure that the algorithm converges to a stationary point. A high quality implementation of such an algorithm is really a minimization method.

Due to enforcement of descent, the iterates will be rolling "down hill", and certainly not likely to terminate (approximately satisfy first order KKT optimality conditions) at a local maximum, unless that is the starting point. And very unlikely to terminate at a saddle point. In general, though, for non-convex problems, such a Newton method may terminate at a local minimum which is not necessarily a global minimum.

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  • $\begingroup$ I'm looking into implementing something like you're describing (a newton-like method that will only converge to local minima) but I'm struggling to find any good references. Any suggestions? $\endgroup$
    – Linus
    Commented Sep 15, 2023 at 7:58
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    $\begingroup$ @!Linus "Numerical Optimization" by Jorge Nocedal , Stephen J. Wright link.springer.com/book/10.1007/978-0-387-40065-5 is a goof book if you want to try implementing such methods. But you need to know the basics of numerical analysis (floating point computation). If you just want to use such a method, you're better off calling an existing high quality implementation. $\endgroup$
    – Mark L. Stone
    Commented Sep 15, 2023 at 10:03
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    $\begingroup$ thanks for the pointer. The algorithm in appendix B (2nd ed) is pretty much what I was looking for $\endgroup$
    – Linus
    Commented Sep 16, 2023 at 11:45

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